Axiom ax-mulass 7852

Description: Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axmulass 7810. Proofs should normally use mulass 7880 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Assertion

Ref	Expression
ax-mulass	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Detailed syntax breakdown of Axiom ax-mulass

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		cc 7747	. . . 4 class ℂ
3	1, 2	wcel 2136	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2136	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2136	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 968	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 7754	. . . . 5 class ·
10	1, 4, 9	co 5841	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 5841	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 5841	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 5841	. . 3 class (𝐴 · (𝐵 · 𝐶))
14	11, 13	wceq 1343	. 2 wff ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
15	8, 14	wi 4	1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

This axiom is referenced by:  mulass  7880